Determination of sub-ps lattice dynamics in FeRh thin films

Understanding the ultrashort time scale structural dynamics of the FeRh metamagnetic phase transition is a key element in developing a complete explanation of the mechanism driving the evolution from an antiferromagnetic to ferromagnetic state. Using an X-ray free electron laser we determine, with sub-ps time resolution, the time evolution of the (–101) lattice diffraction peak following excitation using a 35 fs laser pulse. The dynamics at higher laser fluence indicates the existence of a transient lattice state distinct from the high temperature ferromagnetic phase. By extracting the lattice temperature and comparing it with values obtained in a quasi-static diffraction measurement, we estimate the electron–phonon coupling in FeRh thin films as a function of laser excitation fluence. A model is presented which demonstrates that the transient state is paramagnetic and can be reached by a subset of the phonon bands. A complete description of the FeRh structural dynamics requires consideration of coupling strength variation across the phonon frequencies.


Experimental parameters
The pump-probe x-ray scattering experiment was conducted at BL3 of SACLA, Japan research facility. The scattering geometry, shown in the main text (Fig. 2a), was optimized to achieve similar laser and x-ray probe depths 1 . A nitrogen cryoblower was used to provide a stable temperature, minimising drift, over the period of the experiment. This also allowed us to change the ambient temperature of the sample to determine the effective temperature change that the laser pulse induces for a given fluence. The beamline was operated without monochromator 2 ( E/E ~ 6e -3 ), delivering photons with energies close to 6.408 keV. The x-ray energy was chosen to be below the Fe K-edge in order to minimize the background signal resulting from the excitation of Fe fluorescence. The instrument parameters for the x-ray diffraction measurements and laser excitation of the sample were:

Estimation of x-ray diffraction parameters as function of pump-probe delay
In order to determine how the (-1 0 1) x-ray diffraction peak position and its FWHM evolve with time, we simulated the expected AF and FM peak intensity as a function of scattering angle. Based on the magnetometry results from Fig. 1c, we determined the ratio of each phase at a given temperature.
The 1 st derivative dM/dT can be fitted with a Gaussian function having a centre equal to TT and a width equal to ΔT. We use this model based on previous experiments in the literature that have shown the transition is first order with a distribution of transition temperatures. The differences in transition temperatures can be seen on a microscopic scale which averages out to a Gaussian distribution across thin film samples 3 .
For the thermal distribution model, we then assume the intensity of the AF and FM peaks are in direct proportion to the ratio of each phase. The peaks were modelled as Voigt functions with FWHM taken from the XRD data. The centre of each (-101) peak was found from the respective lattice constants (aAF = 0.299 nm, aFM = 0.302 nm 4 ) subject to the diffraction condition, = ⋅ . We used the xray energy of the pink beam (ΔE = 40 eV ≈ 0.2°) available at SACLA (6.408 keV) for this calculation.
From the sum of the two peaks, we extract the centre of mass (COM) and FWHM by matching the S3 experimental data to the model. Using an asymptotic heating profile to the final temperature, we were then able to replicate the time traces seen in the lattice dynamics in Fig. S1. This simulation confirms the assumption that the transition we observed was first order and the change in Bragg peak can be attributed to phonon heating of FeRh.

Extracting Debye-Waller factor from the static XRD measurements
In order to quantify the lattice expansion based on quasi-static XRD data, we required accurate estimates of the 2θ centre and peak FWHM. As the peaks do not show simple Voigt or Lorentzian shapes, a new model was devised. Following the work of Barton et. Al. 5 , an inhomogeneous strain across the film was proposed to explain the asymmetric peak observed. In this work a shift in the 2θ angle of FeRh (002) is observed as the strain is relived. The data has a tail that appears to have a Voigt shape, with the leading edge composed of a series of strained contributions. This corresponds with a portion of the film being strained at the MgO interface 5 . Then a large portion of the film is unstrained yielding the tail of the peak, it should be noted that quasi-static XRD measurements probe the entire film thickness. To demonstrate the effect of inhomogeneous strain on the peak shape, Fig. S2 illustrates the expected peak shape for a film with a range of lattice constants, with increased contribution for the unstrained portions. This model could be adapted by changing n, the number of peaks with independent lattice constants. As n increases it approaches the behaviour of the continuum of strain as expected for the films presented in this work. The peaks in the Fig. S2a&b show the examples of the peaks for n=2 (solely AF and FM peaks), and the summation of the peaks for n=10 (strain profile).

S4
In order to consistently treat the data, we only fitted to the tail of the data using a region with intensities 80% of the maximum and above as seen in Fig. S2c. This choice of cut-off was arbitrary but is seen to describe the tail end of the peak and therefore the unstrained film. This portion of the film is the main contributor to the peaks observed in the synchrotron experiment due to the grazing incidence geometry. Subject to this cut-off in 2θ, the static XRD peaks were fitted with Voigt functions in order to extract the position, width, and intensity. This allowed the lattice constant of the FeRh at each temperature to be determined. By comparing the expansion to the RT value, the relative expansion was found. The substrate behaviour was found by fitting the MgO (002) peaks and used as a reference for the induced expansion. In order to plot Fig. 3b presented in the main paper, the following relation was used Where δV(T) is the change in volume (proportional to lattice constant), and Vi(T) is the extracted volume at a given temperature from the (002) peak for the respective species. Using the values of integrated intensity extracted from the Voigt fit, we can estimate the Debye temperature of the S5 material and infer the lattice temperatures as a function of time 6 shown in the main paper, Fig. 7a.
The Debye-Waller factor (DWF) presented in Eq. (1) & (2) of the main paper can be rearranged to provide the following estimate of the lattice temperature as a function of the scattering intensity = 0 − Θ 2 3ℏ 2 2 ln ( Where I(T) is the peak intensity, and the remaining variables are defined as in Eq.
(2) of the main text.
We apply the treatment to the heated XRD peaks to find the pre-factor (χD) in the above equation. Fig.   S3 shows the intensity of the static peaks as a function of temperature, using 423 K as T0 as this was the most intense measured peak in Fig. 3a of the main text. The fitted line provides an estimate of the DWF for the FeRh XRD peaks. Using this value of the DWF, we could plot the lattice temperature as a function of time as in the main text, Fig. 7. The pre-factor is changed to account for differences in the estimated average displacement for the (-101) peak. The same treatment is plotted for both different laser fluences and initial temperatures. This analysis provides a ΘD = 234 ± 70 K, which is lower than that found in specific heat measurements. This is due to the Debye model of x-ray intensity only considering acoustic phonons in monatomic systems 7 , so called ΘM. By accounting for the crystal structure and species of FeRh, we predict an adjusted ΘD = 331 ± 80 K, which agrees with ab initio predictions for the FM phase of 300 K 8 .

Extension of the three-temperature model -transient phonon channel
The fitting the short (ps) time dependent diffraction data (Fig. 6) suggests a short-lived state in the phonon bands. By adding a term to the three-temperature model in the form of a new short-lived excited state the kinetics of the system can be explained. In the case of the higher laser excitation scans i.e., those above the 5.5 mJ cm -2 threshold fluence, this pathway becomes a significant part of the transition behaviour. To describe this behaviour, we first built a model based on solving differential equations from the three-temperature assumptions. In this model, the spin coupling is presumed to be relatively weak over the measured timescales. Due to the relatively long growth lifetimes of the magnetisation 9,10 , the kinetics of the electron and phonon systems can be approximated as Where the subscript 'el' and 'l' refer to the electronic and lattice systems, respectively. C refers to the respective heat capacity and γ is the coupling strength. The change in system temperature as a function of time was numerically solved using an ordinary differential equation solver. The value for γ is estimated as the inverse of the extracted growth lifetime τG from Eq. (3) in the main text, and the heat capacity used was 3.20x10 17 J m -3 s -1 K -1 . The heating of the sample from the laser is assumed to be due to the absorbed photons (non-reflected). The skin depth (δp), and the reflectance of the laser light are required to simulate the heating of the electronic system. These are found from the refractive index of FeRh at 800 nm 11 , giving a value of δp=31 nm; while R=0.7 is found from reflectivity data 12 .
In order to model the new highly excited system we adapted the equations to include the excited state by allowing the electronic system to relax via an intermediate channel 6 .
Where '*' refers to the transient lattice state. The laser heating parameters are captured in the first term of the electronic differential equation; I(t) being the laser pulse intensity, R is the reflectivity, and δP is the skin depth and α is to the relative 'hot phonon' population when compared to the entire phonon system. The coupling between the electron and hot phonons is captured using γ*, while coupling between excited and equilibrium states are implemented with γ 1 . These can be estimated from the lifetimes τG and τG* extracted from the fitting performed in the main text, Fig. 6a.

S7
Using previously published data on the calorimetric measurements on FeRh, we can plot the heat capacity of the electron and phonon systems of FeRh 13,14 . The latent heat of the transition is captured as a Gaussian function about the midpoint of the transition (355 K), with a value of 2.2 kJ kg - 1 13,15 . The non-equilibrium coupling between the electron and phonon systems is available for metals including Fe, but has been shown to vary by less than 5% over the temperature range 16 , so a constant value was used. The parameters used in the model are presented in Fig. S4. This model can be adapted to consider the ambient temperature. The effect of this is demonstrated in Fig. S5. It can be seen that the dynamics are slower at higher temperatures and fluences, as we have previously observed when fitting the transient intensity of the peaks. This can be understood by considering that the sample is driven further out of equilibrium when more energy is present in the system either in the form of thermal or laser excitation. This predicts that the peak shift is diminished for the lower ambient temperatures due to the reduced portion of film that enters the FM phase.
Overall, this model slightly overestimates the lattice heating at the lowest fluences and underestimates the high fluence lattice heating; with estimates of the maximum temperature reached S8 being 425, 537, 602 and 683 K for excitations of (2.9, 5.5, 7.1, 9.4 mJ cm -2 ) for the simulations shown in Fig. 7b of the main text. We assume there are other forms of losses not accounted for in the model as we estimated that the lattice heating as a function of laser fluence is 40 K mJ -1 cm 2 , slightly higher than previously reported values in the literature using similar pump laser regimes 9 . There may be other losses not considered as the model assumes any non-reflected light is absorbed by the system.
With regard to the lowest fluence case, the model appears to be underestimating the latent heat for the AF → FM transition. This would explain why the model more closely predicts the behaviour when the laser fluences heat the material above the TT (> 5 mJ cm -2 ). The model could be improved by increasing the heat capacity around TT which could be verified by calorimetric measurements on samples with a range of TT values which has not been explored experimentally to date. The high fluence cases can be explained by considering the intensity, which was assumed to be a direct probe of the lattice temperature. At the highest fluences, the intensity is further decreased by the shift in 2θ caused by the lattice expansion meaning the extracted lattice temperature is slightly increased. In general, this model predicts slightly faster than observed changes of peak intensity and higher equilibrium temperatures for both reduced fluences and lower ambient temperatures. However, this is a good first approximation for the non-trivial electron-phonon coupling as the approach used is mainly based on theoretical studies of the FeRh.